Exponentials: A Story About Chess, The Pandemic, And Technology

A story

I have heard and read many variations of this story. It’s probably not historically accurate, but I still find it fascinating. Once upon a time, the man who invented the game of chess decided to present it to a king who was inconsolable after the loss of his son. The king was captivated by the game and insisted on rewarding the man by granting him one wish.

“I am a simple man, my King. All I want is one grain of rice for the first square on the chessboard, two for the second square, four for the third, eight for the fourth, and so on, until the sixty-fourth square.”

The king laughed and ordered the treasurer to grant the man his wish.

There are two endings to this story. But before we get there, I would like to ask you to take a few seconds to guess how much rice was needed: a bag? fifty bags? a thousand bags?

The pandemic

The pattern of doubling quantities at fixed intervals is well known in mathematics. We refer to this as “exponential growth.” It is observed in several fields: computer science (more about that later), finance (compound interest), physics (chain reactions), and… epidemiology. In fact, when studying a pandemic, exponential growth is the biggest worry: if a virus carrier infects two people, on average, in a week, those two could infect four, who could then infect eight, and so on.

How many people would be infected after four, ten, or sixty-four weeks?

This is a hard calculation for the human brain to make. In fact, many studies conducted since the 1970s have demonstrated that we humans massively underestimate exponential growth and try to compensate for it with something we know much better: “linear growth.”

Linear growth is adding a fixed amount in each step, such as moving from one to three to five. At the tenth step, the linear count is 21, whereas the exponential increase is 1 024!

The difficulty in understanding this difference is what makes managing a pandemic so challenging. When a city’s hospital is at 25% capacity and cases grow at 7% per day, it might seem safe, whereas the system is on the path to be overwhelmed in three weeks. And this will be even more dramatic, because a lockdown will take one to two weeks to have an effect.

Our brains are not wired for exponentials. If you want to see for yourself, let’s go back to our story. How much rice did the chess inventor ask for? What was your guess? Whatever it was, I will take my chances and ask you to go higher. What is your new guess?

The correct answer is roughly equivalent to one thousand times the annual rice production today. In one variation of the story, the man become the king’s treasurer, while in the other he is severely punished for this little trick.

Technology

If you are like most of us, your guess probably came in low. And to reassure you, this doesn’t change if you ask epidemiologists, mathematicians, or programmers (without a two-way brain-computer interface). Nor does it change when you talk about a phenomenon that takes place over a much longer timeline: technology.

Technological evolution is yet another illustration of exponential growth. Over the past few decades, for example, a computer’s processing power, storage capacity, and communication speed have doubled every one to two years. This is the foundational trend behind many of the digital innovations we have witnessed in the last several years: smartphones, cloud computing, blockchain, deep learning, etc.

Where will this evolution take us in five, ten, or twenty years? Whatever your answer, remember that your first guess—and perhaps even your second—likely underestimates actual growth. We all remember how Steve Ballmer underestimated Apple’s entry into the phone business or how Blockbuster turned down Reed Hastings’ offer to sell Netflix for $50 million.

Now that we have the right mindset, I will leave you with another question: how do you think technology will change your home, your business, and your environment in the next five to ten years?